The application of the Petrov-Galerkin projection method in Cosserat rod finite element formulations offers significant advantages in simplifying the expressions within the discrete virtual work functionals. Moreover, it enables a straight-forward and systematic exchange of the ansatz functions, specifically for centerline positions and cross-section orientations. In this concise communication, we present a total Lagrangian finite element formulation for Cosserat rods that attempts to come up with the least required concepts. The chosen discretization preserves objectivity and allows for large displacements/rotations and for large strains. The orientation parametrization with non-unit quaternions results in a singularity-free formulation.

翻译：Petrov-Galerkin投影方法在Cosserat杆有限元公式中的应用，在简化离散虚功泛函表达式方面具有显著优势。此外，该方法能够直接且系统地替换形函数，特别是针对中心线位置和横截面方向。在这篇简明短文中，我们提出了一种总拉格朗日Cosserat杆有限元公式，该公式力求采用最少必要概念。所选离散化方案保持客观性，允许大位移/大转动和大应变。采用非单位四元数进行方向参数化，形成了无奇异性的公式。